Conic neighborhoods ⇔ Polyhedral - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:40:04Z http://mathoverflow.net/feeds/question/23439 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23439/conic-neighborhoods-polyhedral Conic neighborhoods ⇔ Polyhedral Nina Lebedeva 2010-05-04T15:00:25Z 2010-05-04T15:00:25Z <p>I am looking for a reference to the following fact (I can prove it my-self, but it should be known for a century).</p> <blockquote> <p>Let $X$ be a <em>reasonable</em> metric space such that each point has a spherical neighborhood which is isometric to a cone. Then $X$ is a polyhedral space.</p> </blockquote> <p>Reasonable means say <em>compact</em> plus <em>finite Hausdorff dimension</em> (I would be happy with anything which includes finite dimensional Alexandrov space).</p> <p><strong>Definitions:</strong></p> <ul> <li>A finite simplicial complex $P$ with a metric is called <em>polyhedral space</em> if each simplex in $P$ is isometric to a flat simplex.</li> <li>A space $K$ is called <em>cone</em> if there is a metric space $\Sigma$ and $r>0$ such that $K$ is isometric to $\Sigma\times[0,r]$ with metric defined by the law of cosines; i.e. $$|(\xi,x)(\zeta,z)|^2=x^2+y^2-2xy\cos\alpha,$$ where $\alpha$ is the distance from $\xi$ to $\zeta$ in $\Sigma$.</li> </ul>